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## Revista Brasileira de Ensino de Física

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*versão impressa* ISSN 1806-1117*versão On-line* ISSN 1806-9126

### Rev. Bras. Ensino Fís. v.29 n.3 São Paulo 2007

#### http://dx.doi.org/10.1590/S1806-11172007000300008

**ARTIGOS GERAIS**

**A very simple way to measure coaxial cable impedance**

**Um modo simples para medir a impedância de um cabo coaxial**

**P. Fonseca; A.C.F. Santos ^{1}; E.C. Montenegro**

Instituto de Física, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

**ABSTRACT**

A very simple experiment is described to measure the impedance of a coaxial cable and the velocity of propagation of a pulse. The experiment uses a fast pulse, which has the advantage of avoiding long cables and it allows the observation of pulse dispersion.

**Keywords:** coaxial cable, impedance matching, signal transmission.

**RESUMO**

Descrevemos um experimento simples para medir a impedância de um cabo coaxial e a velocidade de propagação de um pulso. O experimento utiliza um pulso rápido, que possui a vantagem de evitar cabos muito longos e permite a observação da dispersão do pulso.

**Palavras-chave:** cabo coaxial, casamento de impedâncias, transmissão de sinais.

**1. Introduction**

Coaxial cables are the interconnections that transmit pulses from one end to another, preserving the information in the signal. Their basic geometry is that of two concentric cylindrical conductors separated by a dielectric material. Coaxial cables are standard transmission lines in nuclear, atomic and high-energy electronics.

Some papers [1-4] have been devoted for demonstrating the properties of transmission lines whether using expensive and sophisticated instruments or low-cost equipment. Most of them use slow pulses (rise time comparable to the transit time in the transmission line). This paper aims to describe a very simple experiment using fast signals that are mostly used in atomic, nuclear and high-energy physics. One advantage of this approach is that it allows ones to use relatively short cables as opposed to cables of hundreds of meters that are often used.

The equivalent circuit for a unit length of transmission line is shown in Fig. 1. For a loss less line *R = G* = 0 and the gradient of voltage and current are given by

and

Differentiating Eqs. (1) and (2) with respect to distance and time, respectively, one obtains

which is the transmission line wave equation. One can easily show by comparing Eq. (3) to the general wave equation that the velocity with which the electromagnetic energy propagates along this loss less line is given by

The speed of signal propagation is more often expressed as its reciprocal, the time of propagation per unit length or delay of the cable, and its typically on the order of 5 ns/m. The characteristic resistance of line is given by

**2. Procedure and results**

The signal in a coaxial cable is, in general, the sum of the incident signal and a reflected signal traveling in the opposite direction. In this section, we describe very simple experiments that make use of some of the transmission lines characteristics, namely, velocity of propagation, reflection and impedance matching.

Pulses from a commercially available generator ORTEK are fed to a constant fraction discriminator (CFD). The timing output of the CFD produces fast logic pulses that are seen by a 100 MHz oscilloscope transmitted by means of a 50 W coaxial cable. Figure 2 shows the main and reflected pulses observed using the oscilloscope without impedance matching. Since oscilloscopes are high impedance devices ( » 1 MW), direct entry of a fast signal will lead in an impedance mismatch and a mistaken signal indication.

**2.1. Measuring the impedance of the coaxial cable**

If the line is terminated with a purely resistive load of impedance *R*, pulse will be reflected or transmitted at the end of the line. The transmission coefficient *T* is a function of the load as

where *T* is defined as the ratio between the transmitted (*V _{t}*) and incident (

*V*) voltages

_{o}In the limiting case of infinite load impedance, the transmitted signal is twice the incident signal. On the other hand, if *R* equals to the impedance of the transmission line, *Z _{cable}*, the transmitted signal is equal to the incident one.

By adding a potentiometer in parallel (shunt), as shown in Fig. 3, one is able to measure the transmitted signal as a function of *R*. Figure 4 shows the plot of the transmission coefficient *T* as a function of *R*. Another manner of studying the dependence of *T* as a function of *R* is by linearizing Eq. (6) as follows

So, the plot of *T*^{-1} as a function of 2*R*^{ - 1} should give rise to a straight line with the angular coefficient equals to *Z _{cable}*. The transmission coefficient is determined as follows: Firstly, one measures the amplitude

*V*

_{t}_{¥}of the fast pulse at the end of the coaxial cable using an oscilloscope without impedance matching as shown in Fig. 1. The pulse of interest is the very first one (in time). The following pulses are related to multiple reflections at both ends of the coaxial cable. According to Eq. (6), the measured signal corresponds to twice the incident signal, i.e.,

*V*

_{t}_{¥}= 2

*V*. Then, one adds the potentiometer in shunt, and measures the amplitude of the transmitted pulse,

_{o}*V*(again the very first one in time) as a function of

_{t}*R*. The transmission coefficient is determined as (see Eq. (7))

Figure 4 shows this plot. The experimental value for *Z _{cable}* is 48 ± 4W, in agreement with the nominal value from the coaxial cable impedance.

**2.2. Velocity of propagation**

One advantage of using fast signal is the fact that it is not necessary to use very long cables. From Fig. 2, one can measure the distance in time between adjacent pulses corresponding to a cable of 3.49 ± 0.01 m which allow us to obtain the velocity of propagation of the pulse as

**2.3. Pulse dispersion and the lossy cable**

From Fig. 2 one can see that width of the reflected pulses increase with the number of reflections. This dispersion of the pulse packet is due to a differential attenuation of the frequency components as a consequence of the fact that the coaxial cable is not ideal, i.e., the velocity depends on the frequency. The general solution of the wave equation for a lossy cable is

where

which represents two traveling waves. The lossy cable leads to an exponential attenuation of the signal with the distance. The attenuation coefficient is often so small that it is significant only in very long cables of many ten's of meters. The dependence of the phase velocity v = *dk/d*w with the frequency is responsible for the dispersion seen in Fig. 2. For coaxial cables, frequencies above 100 MHz are attenuated more efficiently.

**3. Conclusions**

We described a very simple experiment to measure the impedance and velocity of pulse propagation of a coaxial cable using a potentiometer and a non-expensive oscilloscope. Besides the low cost, one additional advantage is the use of not very long cables.

**References**

[1] F. Holuj, Am. J. Phys. **50**, 282 (1982). [ Links ]

[2] M.C. Whatley, Am. J. Phys. **58**, 181 (1990). [ Links ]

[3] G.H. Watson, Am. J. Phys. **63**, 423 (1995). [ Links ]

[4] J.M. Serra, M.C. Brito, J.M. Alves and A.M. Vallera, Eur. J. Phys. **25**, 581 (2004). [ Links ]

[5] W.R. Leo, *Techniques for Nuclear and Particle Physics Experiments* (Springer-Verlag, Berlin, 1987). [ Links ]

Recebido em 14/3/2007; Aceito em 27/4/2007

1 E-mail: toni@if.ufrj.br.